machine learning control – taming nonlinear dynamics and...

Thomas DuriezSteven L. BruntonBernd R. Noack

Machine Learning Control –Taming Nonlinear Dynamics andTurbulence

Springer

Chapter 1Introduction

“I think it’s very important to have a feedback loop, where you’re constantly think-ing about what you’ve done and how you could be doing it better."

- Elon Musk

1.1 Feedback in engineering and living systems

Feedback processes are critical aspects of most living and engineering systems.Feedback occurs when the output of a system influences the input of the same sys-tem. Feedback control is a process of creating such a feedback loop to modify thebehavior of a dynamical system through actuation that is informed by measurementsof the system.

The very existence of humans and other endothermic animals is based on a ro-bust feedback control: They maintain their body temperature within narrow limitsdespite a large range of environmental conditions and disturbances. This tempera-ture regulation is performed with temperature monitoring and control actions, suchas increasing metabolism or sweating. Similarly, air conditioning also keeps a roomtemperature in a narrow interval by heating or cooling via a ventilating air stream.

The world around us is actively shaped by feedback processes, from the mean-dering path of a river to the gene regulation that occurs inside every cell in our body.A child’s education may be considered a feedback control task, where parental andsocietal feedback guide the child’s actions towards a desired goal, such as sociallyacceptable behavior and the child becoming a productive member of society. Theorder achieved in a modern society is the result of a balance of interests regulatedthrough active policing and the rule of laws, which are in turn shaped by a collec-tive sense of justice and civil rights. Financial markets and portfolio managementare also feedback processes based on a control logic of buying and selling stocksto reach an optimal growth or profit at a given risk over a certain time horizon. Infact, currency inflation is actively manipulated by changing interest rates and issuingbonds. Our very thoughts and actions are intimately related to a massively parallelfeedback architecture in our brain and nervous system, whereby external stimuli

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Machine Learning Control – Taming Nonlinear Dynamics and Turbulence

are collected and assimilated, decisions are made, and control actions are executed,resulting in our interaction with the world. Finally, the earth’s climate and temper-ature are maintained through a delicate balance of forcing from sources includingsolar irradiance, greenhouse gases, vegetation, aerosols and cloud formation, manyof which are coupled through feedback.

The feedback control of fluid systems is an immensely important challenge withprofound implications for technologies in energy, security, transportation, medicine,and many other endeavors. Flow control is an academically exciting research fieldundergoing rapid progress — comprising many disciplines, including theoretical,numerical and experimental fluid mechanics, control theory, reduced-order model-ing, nonlinear dynamics and machine learning techniques. Flow control has appli-cations of epic proportion, such as drag reductions of cars, trucks, trains, ships andsubmarines, lift increase of airplanes, noise reduction of ground or airborne trans-port vehicles, combustion efficiency and NOX reduction, cardiac monitoring andintervention, optimization of pharmaceutical and chemical processes and weathercontrol. The flows found in most engineering applications are turbulent, introducingthe complexities of high-dimensionality, multi-scale structures, strong nonlinearitiesand frequency crosstalk as additional challenges.

Feedback turbulence control shares a significant overlap with the other feedbacksystems described above, in the sense that

• the control goal can be defined in mathematical terms;• the control actions are also in a well-defined set;• the unforced system has its own internal chaotic nonlinear dynamics, where

neighboring states may rapidly diverge to different behaviors within the predic-tion horizon;

• the full state is only partially accessible by limited sensors;• there is an underlying evolution equation (i.e., the Navier-Stokes equation) which

provides a high-fidelity description of the system, but may not be useful for con-trol decisions in a real-life experiment.

The last three properties are a generic consequence of high-dimensional nonlineardynamics. However, unlike many of the systems described above, turbulence controlis more benign, as the system quickly forgets its past treatment and the controlexperiments tend to be more reproducible. In other words, the unforced and forcedsystems have a statistical stationarity, i.e. statistical quantities like mean values andvariances are well defined. Regardless, feedback turbulence control is significantlymore complex than most academic control theory tasks, such as stabilization of aninverted pendulum. Hence, improving feedback control architectures that work forturbulence control may have significant impact in other complex systems.

Nature offers compelling examples of feedback flow control that may provideinspiration for engineering efforts. For example, eagles are expert flyers, capable ofrising on thermals or landing gently on a rock or tree despite strong wind gust pertur-bations and other challenging weather conditions. These maneuvers require activefeedback control by sensing the current position and velocity and dynamically ad-justing the control actions involving the motion of wings and feathers. An eagle’s

2 Duriez, Brunton, & Noack


flight is robust to significant uncertainty in the environment and flight conditions,including harsh weather and significant changes to its own body, including mass,geometry, and wing shape. It is unlikely that eagles, or other flying animals, suchas birds, bats, or insects, are operating based on a high-fidelity model of the under-lying Navier-Stokes equations that govern fluid flow. Instead, it is more likely thatthese animals have adapted and learned how to sense and modify dominant coher-ent structures in the fluid that are most responsible for generating forces relevant forflight. Airplanes similarly move on prescribed trajectories at predetermined speedsunder varying wind and weather conditions by adjusting their control surfaces, suchas flaps and ailerons, and engine thrust. However, there is still a tremendous oppor-tunity to improve engineering flight performance using bio-inspired techniques.

This book outlines the use machine learning to design control laws, partiallyinspired by how animals learn control in new environments. This machine learningcontrol (MLC) provides a powerful new framework to control complex dynamicalsystems that are currently beyond the capability of existing methods in control.

1.2 Benefits of feedback control

Figure 1.1 illustrates a general feedback control system. The physical system, alsocalled the plant, is depicted in the blue box. The system is monitored by sensors sand manipulated by actuators b through a control logic depicted in the yellow box.Moreover, the plant is subjected to sensor noise and exogenous disturbances w andthe control shall be optimized with respect to a cost function J.

Physicalsystem

Disturbancew

Actuatorsb

CostJ

Controllaw

Sensorss

Fig. 1.1 General optimization framework for feedback control. The behavior of the physical sys-tem is modified by actuators (inputs, b) through a control law informed by sensor measurementsof the system (outputs, s). The control logic is designed to shape the closed-loop response from theexogenous disturbances w to a high-level objective encoded by the cost function J.

Springer, Fluid Mechanics and Its Applications, 116, 2016 3


Controllaw

Physicalsystem

wr b + s

wd wn

+

Fig. 1.2 Open-loop control diagram. A reference signal wr is fed directly into an open-loop con-troller which specifies a pre-determined actuation signal b. External disturbances (wd) and sensornoise (wn), as well as un-modeled system dynamics and uncertainty, degrade the overall perfor-mance.

One example of an optimization tasks is drag reduction. A physically meaningfuloptimization problem penalizes the actuation. A well-posed drag reduction problemrequests a minimization of the power required to overcome drag Jdrag plus the in-vested actuation power Jact, i.e. the net gain J = Jdrag +Jact. Other examples includelift increase, mixing increase and noise reduction. To keep an airplane on a desiredtrajectory, the thrust and lift need to be kept at a well-defined level. Thus, the controltask becomes a reference tracking problem, in which a reference force — or otherquantity — is commanded. In this case, the cost function penalizes the deviationfrom the desired state and the invested actuation level.

In the case of reference tracking, it is natural to first consider the open-loop con-trol architecture shown in Fig. 1.2. In this strategy, the actuation signal b is chosenbased on knowledge of the system to produce the desired output that matches thecommanded reference signal. This is how many toasters work, where the heatingelement is turned on for a fixed amount of time depending on the desired setting.However, open-loop control is fundamentally incapable of stabilizing an unstablesystem, such as an inverted pendulum, as the plant model would have to be knownperfectly without any uncertainty or disturbances. Open-loop control is also inca-pable of adjusting the actuation signal to compensate for disturbances to the system.

Instead of making control decisions purely based on the desired reference, asin open-loop control, it is possible to close the loop by feeding back sensor mea-surements of the system output so that the controller knows whether or not it isachieving the desired goal. This closed-loop feedback control diagram is shown inFig. 1.3. Sensor-based feedback provides a solution to the issues that occur withopen-loop control. It is often possible to stabilize an unstable system with the aidof sensor feedback, whereas it is never possible to stabilize an unstable system inopen-loop. In addition, closed-loop control is able to compensate for external dis-turbances and model uncertainties, both of which should be measured in the sensoroutput.

Summarizing, feedback control is, for instance, necessary for the following tasks:

• Optimize a state or output with respect to a given cost function;



Controllaw

Physicalsystem

wr + ε b + s

−

Feedback signal

wd wn

+

Fig. 1.3 Closed-loop feedback control diagram. The sensor signal s is fed back and subtractedfrom the reference signal wr . The resulting error ε is used by the controller to specify the actuationsignal b. Feedback is generally able to stabilize unstable plant dynamics while effectively rejectingdisturbances wd and attenuating noise wn.

• Stabilize an unstable system;• Attenuate sensor noise;• Compensate for exogenous disturbances and model uncertainty.

Mathematical formulation of feedback control task

There is a powerful theory of feedback control based on dynamical systems. In thisframework, the plant is modeled by an input–output system:

ddt

a = F(a,b,wd), (1.1a)

s = G(a,b,wn), (1.1b)

consisting of a coupled system of possibly nonlinear differential equations in a statevariable a∈RNa , where Na is the dimension of the state. The actuation input is givenby the vector b∈RNb and this input directly affects the state dynamics in Eq. (1.1a),along with exogenous disturbances wd . The sensor measurements are given by theoutput vector s ∈ RNs , and these measurements may be nonlinear functions of thestate a, the control b and noise wn.

The control task is generally to construct a controller

b = K(s,wr), (1.2)

so that the closed-loop system has desirable properties in terms of stability, attenua-tion of noise, rejection of disturbances, and good reference tracking characteristics.The commanded reference signal is wr. These factors are encoded in the cost func-tion J, which is generally a function of the sensor output, the actuation input, andthe various external signals wr, wd , and wn.



With a well-designed sensor-based feedback control law, it is often possible toobtain a closed-loop system that performs optimally with respect to the chosen costfunction and is robust to model uncertainty, external disturbances, and sensor noise.In fact, most modern control problems are posed in terms of optimization via costminimization. The perspective taken in this book is that machine learning providesa powerful new set of techniques to obtain high-performance control laws even forextremely complicated systems with non-convex cost functions.

1.3 Challenges of feedback control

Most textbooks start with simple feedback control problems. An airplane, for in-stance, may need to keep a certain ground speed. The airplane has a steady-statemap (model) indicating the required thrust (actuation) under ambient flow condi-tion and for an average airplane. Thus, the right thrust may be commanded in anopen-loop manner based on the model, as illustrated in Fig. 1.2.

Yet, each airplane has its own steady-state map and an aging process (model un-certainty). Moreover, the wind (exogenous disturbance) may change the ground ve-locity. Model uncertainty and disturbances require a feedback element: The groundspeed needs to be measured (tachometer) and the thrust needs to be adjusted. If theground speed is too low (high), the thrust needs to be increased (decreased). Thegeneral feedback scheme is illustrated in Fig. 1.3.

Evidently, the control design is simple. There is a single state variable a (speed)which is sensed s (tachometer) and acted upon b (thrust) in a highly predictablemanner and with negligible time delay. We refer to the excellent textbook of Åström& Murray [222] for the applicable control design.

The stabilization of steady solutions to the equations for laminar or transitionalflows requires more refined methods.Navier-Stokes equations A sufficiently de-tailed discretized of the Navier-Stokes equation results in a system with a high-dimensional state, making it computationally expensive to design and implementcontrollers. In addition, time-scales may be very small in real-world fluid applica-tions, such as flow over a wing or in a combustor, making controllers very sensitiveto time delays; these time-delays may be due to sensor and actuator hardware or thecomputational overhead of enacting a control law. Sensor and actuator placement isalso a challenge in high-dimensional fluid systems, with competing goals of decreas-ing time delays and increasing downstream prediction. Finally, many fluid systemsare characterized by strongly non-normal linearized dynamics, meaning that the lin-earized Navier-Stokes equations have nearly parallel eigenvectors resulting in largetransient growth of these modes in response to excitation [67, 262].

Despite inherent nonlinearity, stabilizing a steady state brings the system closerto the equilibrium solution where linearization is increasingly valid. Thus, the fluiddynamics literature contains a rich set of success stories based on linear controlmethods. Examples include the stabilization of the cylinder wake [226, 115, 218,65], of the cavity flow [231], of the boundary layer [172, 11], and of the channel



flow [29], just to name a few [43]. The linear quadratic regulator (LQR) and lin-ear quadratic Gaussian (LQG) are among the most widely used methods for controlbased on computational fluid mechanics. The model-based control of experimen-tal plants requires reduced-order models for computationally tractable on-line deci-sions. For details, we refer to excellent reviews on the applications of linear controltheory in fluids mechanics [231, 161, 13, 250]. The associated reduced-order mod-eling efforts are summarized in these reviews and elaborated in [197, 138].

Optimization of turbulent flows tends to be much more complex. In addition tothe challenges outlined above, the system is strongly nonlinear and is sufficientlyfar from a fixed point or limit cycle that linearization is not typically useful. Thenonlinearity manifests in frequency crosstalk, where actuation at a given frequencymay excite or suppress entirely different frequencies. Fully turbulent dynamics aretypically chaotic and evolve on a high-dimensional attractor, with the dimensionof the attractor generally increasing with the turbulence intensity. These are math-ematical issues in turbulence control, but there are also more practical engineeringissues. These include the cost of implementing a controller (i.e., actuator and sensorhardware, modifying existing designs, etc.), computational requirements to meet ex-ceedingly short time scales imposed by fast dynamics and small length scales, andachieving the required control authority to meaningfully modify the flow.

As a motivating example, let us assume we want to minimize the aerodynamicdrag of a car with, say, 32 blowing actuators, distributed over all four trailing edgesand the same number of pressure sensors distributed over the car. A control logicfor driving the actuators based on the sensor readings shall help to minimize the ef-fective propolsion power required to overcome drag. This highlights the significantchallenges associated with in-time control:

• High-dimensional state;• Strong nonlinearity;• Time delays.

A direct numerical simulation of a suitably discretized Navier-Stokes equation hasnot been performed for wind-tunnel conditions. Even a simplifying large eddy sim-ulation requires at minimum tens of millions of grid points and still has a narrowlow-frequency bandwidth for actuation. Secondly, the turbulent flow does not re-spond linearly to the actuation, so that there is no superposition principle for actua-tion effects. The changes to the flow caused by two actuators acting simultaneouslyis not given by the sum of the responses of the two actuators acting alone. Moreover,actuating at twice the actuation amplitude does not necessarily lead to twice the out-put. The trend may even be reversed. Thirdly, the effect of actuation is generally notmeasured immediately. It may take hundreds or thousands of characteristic timescales to arrive at the converged actuated state [21, 205]. We refer to our reviewarticle on closed-loop turbulence control [43] for in-depth coverage of employedmethods.



1.4 Feedback turbulence control is a grand challenge problem

A high-dimensionsional state space and nonlinear dynamics do not necessarily im-ply unpredictable features. One liter of an ideal gas, for instance, contains O(1024)molecules that move and collide according to Newton’s laws. Elastic collisions sig-nify strongly nonlinear dynamics, and indeed, the numerical simulation of New-ton’s laws at macro-scale based on molecular dynamics will remain intractable fordecades to come. Yet, statistical averages are well described as an analytically com-putable maximum entropy state. This is the statistical foundation of classical ther-modynamics. In contrast, the search for similar closures of turbulence has eludedany comparable success [198]. One reason is the ubiqituous Kolmogorov turbulencecascade. This cascade connects large-scale energy carrying anisotropic coherentstructures with nearly isotropic small-scale dissipative structures over many ordersof magnitudes in scale [106]. The multi-scale physics of turbulence has eluded allmathematical simplifications. Feynman has concluded that ‘Turbulence is the mostimportant unsolved problem of classical physics.’ In other words: a grand challengeproblem.

Turbulence control can be considered an even harder problem compared to find-ing statistical estimates of the unforced state. The control problem seeks to design asmall O(ε) actuation that brings about a large change in the flow. Many approacheswould require a particularly accurate control-oriented closure. The necessary controlmechanism might be pictured as a Maxwellian demon who changes the statisticalproperties of the system by clever actions. Control theory methods often focus onstabilization of equilibria or trajectories. Turbulence, however, is too far from anyfixed point or meaningful trajectory for the applicability of linearized methods. Inthe words of Andrzej Banaszuk (1999):

‘The control theory of turbulence still needs to be invented.’

1.5 Nature teaches us the control design

In the previous section, a generic control strategy for turbulence has been describedas a grand challenge problem. Yet, an eagle can land on a rock performing impres-sive flight maneuvers without a PhD in fluid mechanics or control theory. Naturehas found another way of control design: learning by trial and error.

It is next to impossible to predict the effect of a control policy in a system suchas turbulence where we scarcely understand the unforced dynamics. However, itmay be comparatively easy to test the effectiveness of a control policy in an exper-iment. It is then possible to evolve the control policy by systematic testing, exploit-ing good control policies and exploring alternative ones. Following these principles,Rechenberg [223] and Schwefel [242] have pioneered evolutionary strategies in de-sign problems of fluid mechanics more than 50 years ago at TU Berlin, Germany.



In the last 5 decades, biologically inspired optimization methods have becomeincreasingly powerful. Fleming & Purshouse [103] summarize:

‘The evolutionary computing (EC) field has its origins in four landmark evolutionary ap-proaches: evolutionary programming (EP) (Fogel, Owens, & Walsh, 1966), evolution strate-gies (ES) (Schwefel, 1965; Rechenberg, 1973), genetic algorithms (GA) (Holland, 1975),and genetic programming (GP) (Koza, 1992).’

EP, GA and GP can be considered regression techniques to find input–outputmaps that minimize a cost function. Control design can also be considered a regres-sion task: find the mapping from sensor signals to actuation commands which op-timizes the goal function. Not surprisingly, evolutionary computing is increasinglyused for complex control tasks. For example, EP is used for programming robot mis-sions [272]. GA are used to find optimal parameters of linear control laws [90, 23].And since almost two decades, GP has been employed to optimize nonlinear con-trol laws [91]. Arguably GP is one of the most powerful regression techniques asit leads to analytical control laws of almost arbitrary form. All evolutionary meth-ods are part of the rapidly evolving field of machine learning. There are many othermachine learning techniques to discover input–output maps, such as decision trees,support vector machines (SVM), and neural networks, to name only a few [279]. Infact, the first example of feedback turbulence control with machine learning meth-ods has employed a neural network [171]. In the remainder of this book, we refer tomachine learning control as a strategy using any of the aforementioned data-drivenregression techniques to discover effective control laws.

1.6 Outline of the book

The outline of the book is as follows. Chapter 2 describes the method of machinelearning control (MLC) in detail. In Chapter 3, linear control theory is presented tobuild intuition and describe the most common control framework. This theoreticalfoundation is not required to understand or implement MLC, but it does motivatethe role of feedback and highlight the importance of dynamic estimation. In Chap-ter 4, MLC is benchmarked against known optimal control design of linear systemswithout and with noise. We show that MLC is capable of reproducing the optimallinear control but outperforms these methods even for weak nonlinearities. In Chap-ter 5 we illustrate MLC for a low-dimensional system with frequency crosstalk. Alarge class of fluid flows are described by such a system. We show that the lin-earized system is uncontrollable while MLC discovers the enabling nonlinearity forstabilization. In Chapter 6, we highlight promising results from MLC applied inreal-world feedback turbulence control experiments. Chapter 7 provides a summaryof best practices, tactics and strategies for implementing MLC in practice. Chapter 8presents concluding remarks with an outlook of future developments of MLC.



1.7 Exercises

Exercise 1–1: Name two examples of feedback control systems in everyday life.Define the inputs and outputs of the system, the underlying system state anddynamics, and describe the objective function. Describe the uncertainties in thesystem and the types of noise and disturbances that are likely experienced.

Exercise 1–2: Consider the following plant model:

s = b.

(a) Design an open-loop controller b = K(wr) to track a reference value wr.(b) Now, imagine that the plant model is actually s = 2b. How much error is there

in the open-loop controller from above if we command a value wr = 10?(c) Instead of open-loop control, implement the following closed-loop controller:

b = 10(wr− s). What is the error in the closed-loop system for the same com-mand wr = 10?

Exercise 1–3: Choose a major industry, such as transportation, energy, health-care, etc., and describe an opportunity that could be enabled by closed-loop con-trol of a turbulent fluid. Estimate the rough order of magnitude impact this wouldhave in terms of efficiency, cost, pollution, lives saved, etc. Now, hypothesizewhy these innovations are not commonplace in these industries?


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